Algebra Examples

$\sqrt[3]{162 x^{c} y^{5}} = 3 x^{2} y (\sqrt[3]{6 y^{d}})$

Step 1

Rewrite the equation as $3 x^{2} y (\sqrt[3]{6 y^{d}}) = \sqrt[3]{162 x^{c} y^{5}}$ .

$3 x^{2} y (\sqrt[3]{6 y^{d}}) = \sqrt[3]{162 x^{c} y^{5}}$

Step 2

Divide each term in $3 x^{2} y (\sqrt[3]{6 y^{d}}) = \sqrt[3]{162 x^{c} y^{5}}$ by $3 x^{2} y$ and simplify.

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Step 2.1

Divide each term in $3 x^{2} y (\sqrt[3]{6 y^{d}}) = \sqrt[3]{162 x^{c} y^{5}}$ by $3 x^{2} y$ .

$\frac{3 x^{2} y \sqrt[3]{6 y^{d}}}{3 x^{2} y} = \frac{\sqrt[3]{162 x^{c} y^{5}}}{3 x^{2} y}$

Step 2.2

Simplify the left side.

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Step 2.2.1

Cancel the common factor of $3$ .

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Step 2.2.1.1

Cancel the common factor.

$\frac{3 x^{2} y \sqrt[3]{6 y^{d}}}{3 x^{2} y} = \frac{\sqrt[3]{162 x^{c} y^{5}}}{3 x^{2} y}$

Step 2.2.1.2

Rewrite the expression.

$\frac{x^{2} y \sqrt[3]{6 y^{d}}}{x^{2} y} = \frac{\sqrt[3]{162 x^{c} y^{5}}}{3 x^{2} y}$

Step 2.2.2

Cancel the common factor of $x^{2}$ .

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Step 2.2.2.1

Cancel the common factor.

$\frac{x^{2} y \sqrt[3]{6 y^{d}}}{x^{2} y} = \frac{\sqrt[3]{162 x^{c} y^{5}}}{3 x^{2} y}$

Step 2.2.2.2

Rewrite the expression.

$\frac{y \sqrt[3]{6 y^{d}}}{y} = \frac{\sqrt[3]{162 x^{c} y^{5}}}{3 x^{2} y}$

Step 2.2.3

Cancel the common factor of $y$ .

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Step 2.2.3.1

Cancel the common factor.

$\frac{y \sqrt[3]{6 y^{d}}}{y} = \frac{\sqrt[3]{162 x^{c} y^{5}}}{3 x^{2} y}$

Step 2.2.3.2

Divide $\sqrt[3]{6 y^{d}}$ by $1$ .

$\sqrt[3]{6 y^{d}} = \frac{\sqrt[3]{162 x^{c} y^{5}}}{3 x^{2} y}$

Step 2.3

Simplify the right side.

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Step 2.3.1

Simplify the numerator.

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Step 2.3.1.1

Rewrite $162 x^{c} y^{5}$ as ${(3 y)}^{3} \cdot (6 x^{c} y^{2})$ .

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Step 2.3.1.1.1

Factor $27$ out of $162$ .

$\sqrt[3]{6 y^{d}} = \frac{\sqrt[3]{27 (6) x^{c} y^{5}}}{3 x^{2} y}$

Step 2.3.1.1.2

Rewrite $27$ as $3^{3}$ .

$\sqrt[3]{6 y^{d}} = \frac{\sqrt[3]{3^{3} \cdot 6 x^{c} y^{5}}}{3 x^{2} y}$

Step 2.3.1.1.3

Factor out $y^{3}$ .

$\sqrt[3]{6 y^{d}} = \frac{\sqrt[3]{3^{3} \cdot 6 x^{c} (y^{3} y^{2})}}{3 x^{2} y}$

Step 2.3.1.1.4

Move $x^{c}$ .

$\sqrt[3]{6 y^{d}} = \frac{\sqrt[3]{3^{3} \cdot 6 y^{3} x^{c} y^{2}}}{3 x^{2} y}$

Step 2.3.1.1.5

Move $6$ .

$\sqrt[3]{6 y^{d}} = \frac{\sqrt[3]{3^{3} y^{3} \cdot 6 x^{c} y^{2}}}{3 x^{2} y}$

Step 2.3.1.1.6

Rewrite $3^{3} y^{3}$ as ${(3 y)}^{3}$ .

$\sqrt[3]{6 y^{d}} = \frac{\sqrt[3]{{(3 y)}^{3} \cdot 6 x^{c} y^{2}}}{3 x^{2} y}$

Step 2.3.1.1.7

Add parentheses.

$\sqrt[3]{6 y^{d}} = \frac{\sqrt[3]{{(3 y)}^{3} \cdot 6 (x^{c} y^{2})}}{3 x^{2} y}$

Step 2.3.1.1.8

Add parentheses.

$\sqrt[3]{6 y^{d}} = \frac{\sqrt[3]{{(3 y)}^{3} \cdot (6 x^{c} y^{2})}}{3 x^{2} y}$

Step 2.3.1.2

Pull terms out from under the radical.

$\sqrt[3]{6 y^{d}} = \frac{3 y \sqrt[3]{6 x^{c} y^{2}}}{3 x^{2} y}$

Step 2.3.2

Reduce the expression by cancelling the common factors.

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Step 2.3.2.1

Cancel the common factor of $3$ .

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Step 2.3.2.1.1

Cancel the common factor.

$\sqrt[3]{6 y^{d}} = \frac{3 y \sqrt[3]{6 x^{c} y^{2}}}{3 x^{2} y}$

Step 2.3.2.1.2

Rewrite the expression.

$\sqrt[3]{6 y^{d}} = \frac{y \sqrt[3]{6 x^{c} y^{2}}}{x^{2} y}$

Step 2.3.2.2

Cancel the common factor of $y$ .

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Step 2.3.2.2.1

Cancel the common factor.

$\sqrt[3]{6 y^{d}} = \frac{y \sqrt[3]{6 x^{c} y^{2}}}{x^{2} y}$

Step 2.3.2.2.2

Rewrite the expression.

$\sqrt[3]{6 y^{d}} = \frac{\sqrt[3]{6 x^{c} y^{2}}}{x^{2}}$

Step 2.3.2.3

Reorder factors in $\frac{\sqrt[3]{6 x^{c} y^{2}}}{x^{2}}$ .

$\sqrt[3]{6 y^{d}} = \frac{\sqrt[3]{6 y^{2} x^{c}}}{x^{2}}$

Step 3

To remove the radical on the left side of the equation, cube both sides of the equation.

${\sqrt[3]{6 y^{d}}}^{3} = {(\frac{\sqrt[3]{6 y^{2} x^{c}}}{x^{2}})}^{3}$

Step 4

Simplify each side of the equation.

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Step 4.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[3]{6 y^{d}}$ as ${(6 y^{d})}^{\frac{1}{3}}$ .

${({(6 y^{d})}^{\frac{1}{3}})}^{3} = {(\frac{\sqrt[3]{6 y^{2} x^{c}}}{x^{2}})}^{3}$

Step 4.2

Simplify the left side.

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Step 4.2.1

Simplify ${({(6 y^{d})}^{\frac{1}{3}})}^{3}$ .

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Step 4.2.1.1

Multiply the exponents in ${({(6 y^{d})}^{\frac{1}{3}})}^{3}$ .

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Step 4.2.1.1.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

${(6 y^{d})}^{\frac{1}{3} \cdot 3} = {(\frac{\sqrt[3]{6 y^{2} x^{c}}}{x^{2}})}^{3}$

Step 4.2.1.1.2

Cancel the common factor of $3$ .

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Step 4.2.1.1.2.1

Cancel the common factor.

${(6 y^{d})}^{\frac{1}{3} \cdot 3} = {(\frac{\sqrt[3]{6 y^{2} x^{c}}}{x^{2}})}^{3}$

Step 4.2.1.1.2.2

Rewrite the expression.

${(6 y^{d})}^{1} = {(\frac{\sqrt[3]{6 y^{2} x^{c}}}{x^{2}})}^{3}$

Step 4.2.1.2

Simplify.

$6 y^{d} = {(\frac{\sqrt[3]{6 y^{2} x^{c}}}{x^{2}})}^{3}$

Step 4.3

Simplify the right side.

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Step 4.3.1

Simplify ${(\frac{\sqrt[3]{6 y^{2} x^{c}}}{x^{2}})}^{3}$ .

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Step 4.3.1.1

Apply the product rule to $\frac{\sqrt[3]{6 y^{2} x^{c}}}{x^{2}}$ .

$6 y^{d} = \frac{{\sqrt[3]{6 y^{2} x^{c}}}^{3}}{{(x^{2})}^{3}}$

Step 4.3.1.2

Rewrite ${\sqrt[3]{6 y^{2} x^{c}}}^{3}$ as $6 y^{2} x^{c}$ .

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Step 4.3.1.2.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[3]{6 y^{2} x^{c}}$ as ${(6 y^{2} x^{c})}^{\frac{1}{3}}$ .

$6 y^{d} = \frac{{({(6 y^{2} x^{c})}^{\frac{1}{3}})}^{3}}{{(x^{2})}^{3}}$

Step 4.3.1.2.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$6 y^{d} = \frac{{(6 y^{2} x^{c})}^{\frac{1}{3} \cdot 3}}{{(x^{2})}^{3}}$

Step 4.3.1.2.3

Combine $\frac{1}{3}$ and $3$ .

$6 y^{d} = \frac{{(6 y^{2} x^{c})}^{\frac{3}{3}}}{{(x^{2})}^{3}}$

Step 4.3.1.2.4

Cancel the common factor of $3$ .

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Step 4.3.1.2.4.1

Cancel the common factor.

$6 y^{d} = \frac{{(6 y^{2} x^{c})}^{\frac{3}{3}}}{{(x^{2})}^{3}}$

Step 4.3.1.2.4.2

Rewrite the expression.

$6 y^{d} = \frac{{(6 y^{2} x^{c})}^{1}}{{(x^{2})}^{3}}$

Step 4.3.1.2.5

Simplify.

$6 y^{d} = \frac{6 y^{2} x^{c}}{{(x^{2})}^{3}}$

Step 4.3.1.3

Multiply the exponents in ${(x^{2})}^{3}$ .

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Step 4.3.1.3.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$6 y^{d} = \frac{6 y^{2} x^{c}}{x^{2 \cdot 3}}$

Step 4.3.1.3.2

Multiply $2$ by $3$ .

$6 y^{d} = \frac{6 y^{2} x^{c}}{x^{6}}$

Step 4.3.1.4

Cancel the common factor of $x^{c}$ and $x^{6}$ .

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Step 4.3.1.4.1

Factor $x^{6}$ out of $6 y^{2} x^{c}$ .

$6 y^{d} = \frac{x^{6} (6 y^{2} x^{c - 6})}{x^{6}}$

Step 4.3.1.4.2

Cancel the common factors.

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Step 4.3.1.4.2.1

Multiply by $1$ .

$6 y^{d} = \frac{x^{6} (6 y^{2} x^{c - 6})}{x^{6} \cdot 1}$

Step 4.3.1.4.2.2

Cancel the common factor.

$6 y^{d} = \frac{x^{6} (6 y^{2} x^{c - 6})}{x^{6} \cdot 1}$

Step 4.3.1.4.2.3

Rewrite the expression.

$6 y^{d} = \frac{6 y^{2} x^{c - 6}}{1}$

Step 4.3.1.4.2.4

Divide $6 y^{2} x^{c - 6}$ by $1$ .

$6 y^{d} = 6 y^{2} x^{c - 6}$

Step 5

Solve for $d$ .

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Step 5.1

Divide each term in $6 y^{d} = 6 y^{2} x^{c - 6}$ by $6$ and simplify.

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Step 5.1.1

Divide each term in $6 y^{d} = 6 y^{2} x^{c - 6}$ by $6$ .

$\frac{6 y^{d}}{6} = \frac{6 y^{2} x^{c - 6}}{6}$

Step 5.1.2

Simplify the left side.

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Step 5.1.2.1

Cancel the common factor of $6$ .

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Step 5.1.2.1.1

Cancel the common factor.

$\frac{6 y^{d}}{6} = \frac{6 y^{2} x^{c - 6}}{6}$

Step 5.1.2.1.2

Divide $y^{d}$ by $1$ .

$y^{d} = \frac{6 y^{2} x^{c - 6}}{6}$

Step 5.1.3

Simplify the right side.

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Step 5.1.3.1

Cancel the common factor of $6$ .

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Step 5.1.3.1.1

Cancel the common factor.

$y^{d} = \frac{6 y^{2} x^{c - 6}}{6}$

Step 5.1.3.1.2

Divide $y^{2} x^{c - 6}$ by $1$ .

$y^{d} = y^{2} x^{c - 6}$

Step 5.2

Take the natural logarithm of both sides of the equation to remove the variable from the exponent.

$\ln (y^{d}) = \ln (y^{2} x^{c - 6})$

Step 5.3

Expand $\ln (y^{d})$ by moving $d$ outside the logarithm.

$d \ln (y) = \ln (y^{2} x^{c - 6})$

Step 5.4

Expand the right side.

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Step 5.4.1

Rewrite $\ln (y^{2} x^{c - 6})$ as $\ln (y^{2}) + \ln (x^{c - 6})$ .

$d \ln (y) = \ln (y^{2}) + \ln (x^{c - 6})$

Step 5.4.2

Expand $\ln (y^{2})$ by moving $2$ outside the logarithm.

$d \ln (y) = 2 \ln (y) + \ln (x^{c - 6})$

Step 5.4.3

Expand $\ln (x^{c - 6})$ by moving $c - 6$ outside the logarithm.

$d \ln (y) = 2 \ln (y) + (c - 6) \ln (x)$

Step 5.5

Simplify the right side.

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Step 5.5.1

Apply the distributive property.

$d \ln (y) = 2 \ln (y) + c \ln (x) - 6 \ln (x)$

Step 5.6

Move all the terms containing a logarithm to the left side of the equation.

$d \ln (y) - 2 \ln (y) - c \ln (x) + 6 \ln (x) = 0$

Step 5.7

Move all terms not containing $d$ to the right side of the equation.

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Step 5.7.1

Add $2 \ln (y)$ to both sides of the equation.

$d \ln (y) - c \ln (x) + 6 \ln (x) = 2 \ln (y)$

Step 5.7.2

Add $c \ln (x)$ to both sides of the equation.

$d \ln (y) + 6 \ln (x) = 2 \ln (y) + c \ln (x)$

Step 5.7.3

Subtract $6 \ln (x)$ from both sides of the equation.

$d \ln (y) = 2 \ln (y) + c \ln (x) - 6 \ln (x)$

Step 5.8

Divide each term in $d \ln (y) = 2 \ln (y) + c \ln (x) - 6 \ln (x)$ by $\ln (y)$ and simplify.

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Step 5.8.1

Divide each term in $d \ln (y) = 2 \ln (y) + c \ln (x) - 6 \ln (x)$ by $\ln (y)$ .

$\frac{d \ln (y)}{\ln (y)} = \frac{2 \ln (y)}{\ln (y)} + \frac{c \ln (x)}{\ln (y)} + \frac{- 6 \ln (x)}{\ln (y)}$

Step 5.8.2

Simplify the left side.

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Step 5.8.2.1

Cancel the common factor of $\ln (y)$ .

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Step 5.8.2.1.1

Cancel the common factor.

$\frac{d \ln (y)}{\ln (y)} = \frac{2 \ln (y)}{\ln (y)} + \frac{c \ln (x)}{\ln (y)} + \frac{- 6 \ln (x)}{\ln (y)}$

Step 5.8.2.1.2

Divide $d$ by $1$ .

$d = \frac{2 \ln (y)}{\ln (y)} + \frac{c \ln (x)}{\ln (y)} + \frac{- 6 \ln (x)}{\ln (y)}$

Step 5.8.3

Simplify the right side.

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Step 5.8.3.1

Simplify each term.

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Step 5.8.3.1.1

Cancel the common factor of $\ln (y)$ .

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Step 5.8.3.1.1.1

Cancel the common factor.

$d = \frac{2 \ln (y)}{\ln (y)} + \frac{c \ln (x)}{\ln (y)} + \frac{- 6 \ln (x)}{\ln (y)}$

Step 5.8.3.1.1.2

Divide $2$ by $1$ .

$d = 2 + \frac{c \ln (x)}{\ln (y)} + \frac{- 6 \ln (x)}{\ln (y)}$

Step 5.8.3.1.2

Move the negative in front of the fraction.

$d = 2 + \frac{c \ln (x)}{\ln (y)} - \frac{6 \ln (x)}{\ln (y)}$